I am trying to learn about homogeneous and symmetric spaces. I know that the quotient $SO(n+1)/SO(n)$ should look like the sphere $S^n$.
But how so (i.e, what is the equivalence criterion)? Is $SO(n)$ a normal subgroup and the quotient has a group isomorphism with $S^n$, or we simply have a diffeomorphism between the coset space $SO(n+1)/SO(n)$ and the sphere $S^n$?
Also, it would be really helpful if somebody can please suggest some survey/introductory literature on homogeneous spaces. Thank you very much for your help, comments are welcome!
Since $S^n$ is not a group for $n\neq 1,3$ (with those cases being the unit complex numbers and the unit quaternions under multiplication) $SO(n)$ is not a normal subgroup of $SO(n+1)$ in general. The group $SO(n)$ is a smooth subspace of the space $\mathbb R^{n^2}$ of $n$ by $n$ real matrices, and so if we put Euclidean topology on $\mathbb R^{n^2}$ then $SO(n)$ inherits the subspace topology. The claim is then that the space of right (say) cosets $SO(n+1)/SO(n)$ with the quotient topology is homeomorphic to $S^n$. One can further put differentiable structures on these spaces (as $SO(n)$ is a Lie group) and upgrade the homeomorphism to a diffeomorphism.
You can see the result from the orbit-stabiliser theorem. $SO(n+1)$ acts on $\mathbb R^{n+1}$ by matrix multiplication. Consider the point $(1,0,\dots,0)$. Its orbit is $S^n$, while its stabiliser is the subgroup $SO(n)$ and so the result follows.
More generally for a Lie group $G$ with $H$ some Lie subgroup the space of right cosets $G/H$ is a homogeneous space. The space of cosets is a group iff $H$ is normal in $G$.